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 A066739 Number of representations of n as a sum of products of positive integers. 1 is not allowed as a factor, unless it is the only factor. Representations which differ only in the order of terms or factors are considered equivalent. 48
 1, 1, 2, 3, 6, 8, 14, 19, 32, 44, 67, 91, 139, 186, 269, 362, 518, 687, 960, 1267, 1747, 2294, 3106, 4052, 5449, 7063, 9365, 12092, 15914, 20422, 26639, 34029, 44091, 56076, 72110, 91306, 116808, 147272, 187224, 235201, 297594, 372390, 468844, 584644, 732942 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Table of n, a(n) for n = 0..10000 N. J. A. Sloane, Transforms FORMULA a(n) = Sum_{pi} Product_{m=1..n} binomial(k(m)+A001055(m)-1, k(m)), where pi runs through all partitions k(1) + 2 * k( 2) + ... + n * k(n) = n. a(n)=1/n*Sum_{m=1..n} a(n-m)*b(m), n > 0, a(0)=1, b(m)=Sum_{d|m} d*A001055(d). Euler transform of A001055(n): Product_{m=1..infinity} (1-x^m)^(-A001055(m)). - Vladeta Jovovic, Jan 21 2002 EXAMPLE For n=5, 5 = 4+1 = 2*2+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1, so a(5) = 8. For n=8, 8 = 4*2 = 2*2*2 = ... = 4+4 = 2*2+4 = 2*2+2*2 = ...; note that there are 3 ways to factor the terms of 4+4. In general, if a partition contains a number k exactly r times, then the number of ways to factor the k's is the binomial coefficient C(A001055(k)+r-1,r). MAPLE with(numtheory): b:= proc(n, k) option remember; `if`(n>k, 0, 1) +`if`(isprime(n), 0, add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n})) end: a:= proc(n) option remember; `if`(n=0, 1, add(add(d*b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=0..60); # Alois P. Heinz, Apr 22 2012 MATHEMATICA p[ n_, 1 ] := If[ n==1, 1, 0 ]; p[ 1, k_ ] := 1; p[ n_, k_ ] := p[ n, k ]=p[ n, k-1 ]+If[ Mod[ n, k ]==0, p[ n/k, k ], 0 ]; A001055[ n_ ] := p[ n, n ]; a[ n_, 1 ] := 1; a[ 0, k_ ] := 1; a[ n_, k_ ] := If[ k>n, a[ n, n ], a[ n, k ]=a[ n, k-1 ]+Sum[ Binomial[ A001055[ k ]+r-1, r ]a[ n-k*r, k-1 ], {r, 1, Floor[ n/k ]} ] ]; a[ n_ ] := a[ n, n ]; (* p[ n, k ]=number of factorizations of n with factors <= k. a[ n, k ]=number of representations of n as a sum of products of positive integers, with summands <= k *) b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]]; a[0] = 1; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[j, #*b[#, #]&]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 10 2015, after Alois P. Heinz *) facs[n_]:=If[n<=1, {{}}, Join@@Table[(Prepend[#1, d]&)/@Select[facs[n/d], Min@@#1>=d&], {d, Rest[Divisors[n]]}]]; Table[Length[Union[Sort/@Join@@Table[Tuples[facs/@ptn], {ptn, IntegerPartitions[n]}]]], {n, 50}] (* Gus Wiseman, Sep 05 2018 *) PROG (Python) from sympy.core.cache import cacheit from sympy import divisors, isprime @cacheit def b(n, k): return (0 if n>k else 1) + (0 if isprime(n) else sum([0 if d>k else b(n//d, d) for d in divisors(n)[1:-1]])) @cacheit def a(n): return 1 if n==0 else sum(sum(d*b(d, d) for d in divisors(j))*a(n - j) for j in range(1, n + 1))//n print([a(n) for n in range(61)]) # Indranil Ghosh, Aug 19 2017, after Maple code CROSSREFS Cf. A000041, A001055. Cf. A066815, A066816, A066806. Cf. A001970, A050336, A063834, A065026, A066739, A281113, A284639, A318948, A318949. Sequence in context: A182269 A321360 A321566 * A066815 A106182 A360621 Adjacent sequences: A066736 A066737 A066738 * A066740 A066741 A066742 KEYWORD easy,nonn AUTHOR Naohiro Nomoto, Jan 16 2002 EXTENSIONS Edited by Dean Hickerson, Jan 19 2002 STATUS approved

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Last modified August 4 01:12 EDT 2024. Contains 374905 sequences. (Running on oeis4.)